\(\renewcommand{\hat}[1]{\widehat{#1}}\)

Shared Qs (u10)


  1. Question

    Personally, I find it difficult to interpret large areas. I prefer to find an approximate length to help imagine the scale. If you approximate the shape as a square, the calculation is pretty quick.

    For example, maybe you are told that the lake below has an area of \(290\) \(\mathrm{miles^2}\).

    plot of chunk unnamed-chunk-1

    If you approximate the shape as a square, what is the edge length? In other words, what is the approximate distance across the lake?


    Solution


  2. Question

    Personally, I find it difficult to interpret large areas. I prefer to find an approximate length to help imagine the scale. If you approximate the shape as a square, the calculation is pretty quick.

    For example, maybe you are told that the lake below has an area of \(140\) \(\mathrm{miles^2}\).

    plot of chunk unnamed-chunk-1

    If you approximate the shape as a square, what is the edge length? In other words, what is the approximate distance across the lake?


    Solution


  3. Question

    Personally, I find it difficult to interpret large areas. I prefer to find an approximate length to help imagine the scale. If you approximate the shape as a square, the calculation is pretty quick.

    For example, maybe you are told that the lake below has an area of \(5900\) \(\mathrm{miles^2}\).

    plot of chunk unnamed-chunk-1

    If you approximate the shape as a square, what is the edge length? In other words, what is the approximate distance across the lake?


    Solution


  4. Question

    Personally, I find it difficult to interpret large areas. I prefer to find an approximate length to help imagine the scale. If you approximate the shape as a square, the calculation is pretty quick.

    For example, maybe you are told that the lake below has an area of \(150\) \(\mathrm{miles^2}\).

    plot of chunk unnamed-chunk-1

    If you approximate the shape as a square, what is the edge length? In other words, what is the approximate distance across the lake?


    Solution


  5. Question

    Personally, I find it difficult to interpret large areas. I prefer to find an approximate length to help imagine the scale. If you approximate the shape as a square, the calculation is pretty quick.

    For example, maybe you are told that the lake below has an area of \(2300\) \(\mathrm{miles^2}\).

    plot of chunk unnamed-chunk-1

    If you approximate the shape as a square, what is the edge length? In other words, what is the approximate distance across the lake?


    Solution


  6. Question

    Personally, I find it difficult to interpret large areas. I prefer to find an approximate length to help imagine the scale. If you approximate the shape as a square, the calculation is pretty quick.

    For example, maybe you are told that the lake below has an area of \(78\) \(\mathrm{miles^2}\).

    plot of chunk unnamed-chunk-1

    If you approximate the shape as a square, what is the edge length? In other words, what is the approximate distance across the lake?


    Solution


  7. Question

    Personally, I find it difficult to interpret large areas. I prefer to find an approximate length to help imagine the scale. If you approximate the shape as a square, the calculation is pretty quick.

    For example, maybe you are told that the lake below has an area of \(1500\) \(\mathrm{miles^2}\).

    plot of chunk unnamed-chunk-1

    If you approximate the shape as a square, what is the edge length? In other words, what is the approximate distance across the lake?


    Solution


  8. Question

    Personally, I find it difficult to interpret large areas. I prefer to find an approximate length to help imagine the scale. If you approximate the shape as a square, the calculation is pretty quick.

    For example, maybe you are told that the lake below has an area of \(760\) \(\mathrm{miles^2}\).

    plot of chunk unnamed-chunk-1

    If you approximate the shape as a square, what is the edge length? In other words, what is the approximate distance across the lake?


    Solution


  9. Question

    Personally, I find it difficult to interpret large areas. I prefer to find an approximate length to help imagine the scale. If you approximate the shape as a square, the calculation is pretty quick.

    For example, maybe you are told that the lake below has an area of \(250\) \(\mathrm{miles^2}\).

    plot of chunk unnamed-chunk-1

    If you approximate the shape as a square, what is the edge length? In other words, what is the approximate distance across the lake?


    Solution


  10. Question

    Personally, I find it difficult to interpret large areas. I prefer to find an approximate length to help imagine the scale. If you approximate the shape as a square, the calculation is pretty quick.

    For example, maybe you are told that the lake below has an area of \(4100\) \(\mathrm{miles^2}\).

    plot of chunk unnamed-chunk-1

    If you approximate the shape as a square, what is the edge length? In other words, what is the approximate distance across the lake?


    Solution


  11. Question

    Consider the \(y=\sqrt{x}\) curve shown below (the square root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt{x}\)
    22.3
    5.51
    54.5
    8.38
    74.6
    9.04


    Solution


  12. Question

    Consider the \(y=\sqrt{x}\) curve shown below (the square root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt{x}\)
    9.5
    5.5
    37.8
    6.78
    58.2
    9.75


    Solution


  13. Question

    Consider the \(y=\sqrt{x}\) curve shown below (the square root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt{x}\)
    25.1
    5.7
    37
    6.7
    76.7
    9.09


    Solution


  14. Question

    Consider the \(y=\sqrt{x}\) curve shown below (the square root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt{x}\)
    8.6
    3.38
    13.3
    4.76
    36.8
    9.06


    Solution


  15. Question

    Consider the \(y=\sqrt{x}\) curve shown below (the square root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt{x}\)
    5
    3.69
    65.4
    8.56
    79.4
    9.41


    Solution


  16. Question

    Consider the \(y=\sqrt{x}\) curve shown below (the square root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt{x}\)
    8.2
    3.61
    24.3
    7.02
    54.8
    7.96


    Solution


  17. Question

    Consider the \(y=\sqrt{x}\) curve shown below (the square root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt{x}\)
    4
    5.8
    55.8
    8.21
    75.2
    8.81


    Solution


  18. Question

    Consider the \(y=\sqrt{x}\) curve shown below (the square root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt{x}\)
    4
    5.26
    29.3
    5.86
    58.1
    9.85


    Solution


  19. Question

    Consider the \(y=\sqrt{x}\) curve shown below (the square root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt{x}\)
    4.2
    3.08
    12.6
    4.75
    44.6
    8.08


    Solution


  20. Question

    Consider the \(y=\sqrt{x}\) curve shown below (the square root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt{x}\)
    19.4
    5.77
    41.6
    6.95
    72.9
    8.6


    Solution


  21. Question

    Consider the \(y=\sqrt{x}\) curve shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 8 graphs are made by translating and reflecting the \(y=\sqrt{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 8 graphs with the equations.

    1. Equation \(y=-\sqrt{-(x+4)}-3\) matches graph .
    2. Equation \(y=\sqrt{-(x-4)}+3\) matches graph .
    3. Equation \(y=-\sqrt{-(x-4)}+3\) matches graph .
    4. Equation \(y=\sqrt{x-4}-3\) matches graph .
    5. Equation \(y=-\sqrt{-(x+4)}+3\) matches graph .
    6. Equation \(y=-\sqrt{x-4}+3\) matches graph .
    7. Equation \(y=\sqrt{-(x+4)}-3\) matches graph .
    8. Equation \(y=\sqrt{x-4}+3\) matches graph .


    Solution


  22. Question

    Consider the \(y=\sqrt{x}\) curve shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 8 graphs are made by translating and reflecting the \(y=\sqrt{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 8 graphs with the equations.

    1. Equation \(y=-\sqrt{-(x-5)}-2\) matches graph .
    2. Equation \(y=\sqrt{x-5}-2\) matches graph .
    3. Equation \(y=-\sqrt{x-5}-2\) matches graph .
    4. Equation \(y=\sqrt{-(x+5)}-2\) matches graph .
    5. Equation \(y=-\sqrt{x+5}+2\) matches graph .
    6. Equation \(y=-\sqrt{-(x-5)}+2\) matches graph .
    7. Equation \(y=-\sqrt{-(x+5)}-2\) matches graph .
    8. Equation \(y=-\sqrt{x+5}-2\) matches graph .


    Solution


  23. Question

    Consider the \(y=\sqrt{x}\) curve shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 8 graphs are made by translating and reflecting the \(y=\sqrt{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 8 graphs with the equations.

    1. Equation \(y=-\sqrt{x-2}+5\) matches graph .
    2. Equation \(y=\sqrt{-(x+2)}+5\) matches graph .
    3. Equation \(y=-\sqrt{-(x-2)}-5\) matches graph .
    4. Equation \(y=\sqrt{x+2}+5\) matches graph .
    5. Equation \(y=-\sqrt{x+2}-5\) matches graph .
    6. Equation \(y=\sqrt{x-2}-5\) matches graph .
    7. Equation \(y=\sqrt{-(x-2)}+5\) matches graph .
    8. Equation \(y=-\sqrt{-(x+2)}+5\) matches graph .


    Solution


  24. Question

    Consider the \(y=\sqrt{x}\) curve shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 8 graphs are made by translating and reflecting the \(y=\sqrt{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 8 graphs with the equations.

    1. Equation \(y=\sqrt{x+2}+5\) matches graph .
    2. Equation \(y=\sqrt{-(x+2)}+5\) matches graph .
    3. Equation \(y=\sqrt{-(x-2)}-5\) matches graph .
    4. Equation \(y=\sqrt{-(x+2)}-5\) matches graph .
    5. Equation \(y=-\sqrt{x+2}+5\) matches graph .
    6. Equation \(y=-\sqrt{x+2}-5\) matches graph .
    7. Equation \(y=-\sqrt{x-2}-5\) matches graph .
    8. Equation \(y=\sqrt{-(x-2)}+5\) matches graph .


    Solution


  25. Question

    Consider the \(y=\sqrt{x}\) curve shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 8 graphs are made by translating and reflecting the \(y=\sqrt{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 8 graphs with the equations.

    1. Equation \(y=\sqrt{x-5}-3\) matches graph .
    2. Equation \(y=-\sqrt{-(x-5)}+3\) matches graph .
    3. Equation \(y=\sqrt{-(x+5)}-3\) matches graph .
    4. Equation \(y=-\sqrt{x+5}+3\) matches graph .
    5. Equation \(y=-\sqrt{x-5}+3\) matches graph .
    6. Equation \(y=\sqrt{x+5}+3\) matches graph .
    7. Equation \(y=\sqrt{-(x+5)}+3\) matches graph .
    8. Equation \(y=\sqrt{-(x-5)}+3\) matches graph .


    Solution


  26. Question

    Consider the \(y=\sqrt{x}\) curve shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 8 graphs are made by translating and reflecting the \(y=\sqrt{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 8 graphs with the equations.

    1. Equation \(y=\sqrt{-(x+5)}-3\) matches graph .
    2. Equation \(y=-\sqrt{-(x+5)}+3\) matches graph .
    3. Equation \(y=-\sqrt{-(x-5)}-3\) matches graph .
    4. Equation \(y=-\sqrt{x+5}+3\) matches graph .
    5. Equation \(y=\sqrt{x+5}+3\) matches graph .
    6. Equation \(y=\sqrt{x-5}+3\) matches graph .
    7. Equation \(y=-\sqrt{-(x+5)}-3\) matches graph .
    8. Equation \(y=\sqrt{-(x-5)}-3\) matches graph .


    Solution


  27. Question

    Consider the \(y=\sqrt{x}\) curve shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 8 graphs are made by translating and reflecting the \(y=\sqrt{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 8 graphs with the equations.

    1. Equation \(y=\sqrt{x+4}-3\) matches graph .
    2. Equation \(y=\sqrt{x+4}+3\) matches graph .
    3. Equation \(y=-\sqrt{x+4}-3\) matches graph .
    4. Equation \(y=-\sqrt{x-4}-3\) matches graph .
    5. Equation \(y=\sqrt{-(x+4)}-3\) matches graph .
    6. Equation \(y=\sqrt{-(x-4)}-3\) matches graph .
    7. Equation \(y=-\sqrt{-(x+4)}+3\) matches graph .
    8. Equation \(y=\sqrt{x-4}+3\) matches graph .


    Solution


  28. Question

    Consider the \(y=\sqrt{x}\) curve shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 8 graphs are made by translating and reflecting the \(y=\sqrt{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 8 graphs with the equations.

    1. Equation \(y=-\sqrt{-(x+4)}-2\) matches graph .
    2. Equation \(y=\sqrt{-(x-4)}+2\) matches graph .
    3. Equation \(y=-\sqrt{-(x-4)}+2\) matches graph .
    4. Equation \(y=\sqrt{-(x+4)}-2\) matches graph .
    5. Equation \(y=-\sqrt{x-4}+2\) matches graph .
    6. Equation \(y=\sqrt{x-4}-2\) matches graph .
    7. Equation \(y=-\sqrt{x-4}-2\) matches graph .
    8. Equation \(y=\sqrt{x+4}-2\) matches graph .


    Solution


  29. Question

    Consider the \(y=\sqrt{x}\) curve shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 8 graphs are made by translating and reflecting the \(y=\sqrt{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 8 graphs with the equations.

    1. Equation \(y=\sqrt{x+3}+5\) matches graph .
    2. Equation \(y=-\sqrt{-(x+3)}-5\) matches graph .
    3. Equation \(y=-\sqrt{-(x+3)}+5\) matches graph .
    4. Equation \(y=-\sqrt{-(x-3)}+5\) matches graph .
    5. Equation \(y=-\sqrt{x+3}+5\) matches graph .
    6. Equation \(y=\sqrt{-(x-3)}-5\) matches graph .
    7. Equation \(y=-\sqrt{x+3}-5\) matches graph .
    8. Equation \(y=\sqrt{x+3}-5\) matches graph .


    Solution


  30. Question

    Consider the \(y=\sqrt{x}\) curve shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 8 graphs are made by translating and reflecting the \(y=\sqrt{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 8 graphs with the equations.

    1. Equation \(y=\sqrt{x+3}-5\) matches graph .
    2. Equation \(y=-\sqrt{-(x-3)}+5\) matches graph .
    3. Equation \(y=\sqrt{-(x-3)}+5\) matches graph .
    4. Equation \(y=-\sqrt{x-3}-5\) matches graph .
    5. Equation \(y=\sqrt{x-3}-5\) matches graph .
    6. Equation \(y=-\sqrt{-(x+3)}+5\) matches graph .
    7. Equation \(y=\sqrt{-(x-3)}-5\) matches graph .
    8. Equation \(y=-\sqrt{x+3}+5\) matches graph .


    Solution


  31. Question

    The function \(f(x)~=~\sqrt{x}\) has some integer input-output pairs (see Diophantine equations). The first 7 are listed and graphed below.

    \(x\) \(y=f(x)\)
    0 0
    1 1
    4 2
    9 3
    16 4
    25 5
    36 6

    plot of chunk unnamed-chunk-1

    A shifted function \(g\) is defined as \(g(x)=\sqrt{x-4}+2\). Find the corresponding points of integer input and integer output on function \(g\).

    \(x\) \(y=g(x)\)


    Solution


  32. Question

    The function \(f(x)~=~\sqrt{x}\) has some integer input-output pairs (see Diophantine equations). The first 7 are listed and graphed below.

    \(x\) \(y=f(x)\)
    0 0
    1 1
    4 2
    9 3
    16 4
    25 5
    36 6

    plot of chunk unnamed-chunk-1

    A shifted function \(g\) is defined as \(g(x)=\sqrt{x-5}+1\). Find the corresponding points of integer input and integer output on function \(g\).

    \(x\) \(y=g(x)\)


    Solution


  33. Question

    The function \(f(x)~=~\sqrt{x}\) has some integer input-output pairs (see Diophantine equations). The first 7 are listed and graphed below.

    \(x\) \(y=f(x)\)
    0 0
    1 1
    4 2
    9 3
    16 4
    25 5
    36 6

    plot of chunk unnamed-chunk-1

    A shifted function \(g\) is defined as \(g(x)=\sqrt{x-2}+4\). Find the corresponding points of integer input and integer output on function \(g\).

    \(x\) \(y=g(x)\)


    Solution


  34. Question

    The function \(f(x)~=~\sqrt{x}\) has some integer input-output pairs (see Diophantine equations). The first 7 are listed and graphed below.

    \(x\) \(y=f(x)\)
    0 0
    1 1
    4 2
    9 3
    16 4
    25 5
    36 6

    plot of chunk unnamed-chunk-1

    A shifted function \(g\) is defined as \(g(x)=\sqrt{x-5}+1\). Find the corresponding points of integer input and integer output on function \(g\).

    \(x\) \(y=g(x)\)


    Solution


  35. Question

    The function \(f(x)~=~\sqrt{x}\) has some integer input-output pairs (see Diophantine equations). The first 7 are listed and graphed below.

    \(x\) \(y=f(x)\)
    0 0
    1 1
    4 2
    9 3
    16 4
    25 5
    36 6

    plot of chunk unnamed-chunk-1

    A shifted function \(g\) is defined as \(g(x)=\sqrt{x+5}+4\). Find the corresponding points of integer input and integer output on function \(g\).

    \(x\) \(y=g(x)\)


    Solution


  36. Question

    The function \(f(x)~=~\sqrt{x}\) has some integer input-output pairs (see Diophantine equations). The first 7 are listed and graphed below.

    \(x\) \(y=f(x)\)
    0 0
    1 1
    4 2
    9 3
    16 4
    25 5
    36 6

    plot of chunk unnamed-chunk-1

    A shifted function \(g\) is defined as \(g(x)=\sqrt{x+5}-2\). Find the corresponding points of integer input and integer output on function \(g\).

    \(x\) \(y=g(x)\)


    Solution


  37. Question

    The function \(f(x)~=~\sqrt{x}\) has some integer input-output pairs (see Diophantine equations). The first 7 are listed and graphed below.

    \(x\) \(y=f(x)\)
    0 0
    1 1
    4 2
    9 3
    16 4
    25 5
    36 6

    plot of chunk unnamed-chunk-1

    A shifted function \(g\) is defined as \(g(x)=\sqrt{x-3}-5\). Find the corresponding points of integer input and integer output on function \(g\).

    \(x\) \(y=g(x)\)


    Solution


  38. Question

    The function \(f(x)~=~\sqrt{x}\) has some integer input-output pairs (see Diophantine equations). The first 7 are listed and graphed below.

    \(x\) \(y=f(x)\)
    0 0
    1 1
    4 2
    9 3
    16 4
    25 5
    36 6

    plot of chunk unnamed-chunk-1

    A shifted function \(g\) is defined as \(g(x)=\sqrt{x+5}-1\). Find the corresponding points of integer input and integer output on function \(g\).

    \(x\) \(y=g(x)\)


    Solution


  39. Question

    The function \(f(x)~=~\sqrt{x}\) has some integer input-output pairs (see Diophantine equations). The first 7 are listed and graphed below.

    \(x\) \(y=f(x)\)
    0 0
    1 1
    4 2
    9 3
    16 4
    25 5
    36 6

    plot of chunk unnamed-chunk-1

    A shifted function \(g\) is defined as \(g(x)=\sqrt{x-1}-3\). Find the corresponding points of integer input and integer output on function \(g\).

    \(x\) \(y=g(x)\)


    Solution


  40. Question

    The function \(f(x)~=~\sqrt{x}\) has some integer input-output pairs (see Diophantine equations). The first 7 are listed and graphed below.

    \(x\) \(y=f(x)\)
    0 0
    1 1
    4 2
    9 3
    16 4
    25 5
    36 6

    plot of chunk unnamed-chunk-1

    A shifted function \(g\) is defined as \(g(x)=\sqrt{x-3}-4\). Find the corresponding points of integer input and integer output on function \(g\).

    \(x\) \(y=g(x)\)


    Solution


  41. Question

    Consider the \(y=\sqrt[3]{x}\) curve shown below (the cube root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt[3]{x}\)
    -899
    -9.09
    -66
    4.56
    599
    9.33


    Solution


  42. Question

    Consider the \(y=\sqrt[3]{x}\) curve shown below (the cube root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt[3]{x}\)
    -825
    -9.08
    -355
    -4.82
    -35
    4.06


    Solution


  43. Question

    Consider the \(y=\sqrt[3]{x}\) curve shown below (the cube root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt[3]{x}\)
    -375
    -6.6
    -141
    4.14
    161
    8.06


    Solution


  44. Question

    Consider the \(y=\sqrt[3]{x}\) curve shown below (the cube root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt[3]{x}\)
    -769
    -7.67
    -321
    3.39
    432
    8.68


    Solution


  45. Question

    Consider the \(y=\sqrt[3]{x}\) curve shown below (the cube root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt[3]{x}\)
    -910
    -8.48
    -439
    -5.96
    514
    8.54


    Solution


  46. Question

    Consider the \(y=\sqrt[3]{x}\) curve shown below (the cube root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt[3]{x}\)
    -510
    -7.19
    -186
    -4.85
    432
    8.85


    Solution


  47. Question

    Consider the \(y=\sqrt[3]{x}\) curve shown below (the cube root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt[3]{x}\)
    -863
    -9.15
    243
    6.87
    469
    8.6


    Solution


  48. Question

    Consider the \(y=\sqrt[3]{x}\) curve shown below (the cube root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt[3]{x}\)
    -555
    -7.7
    8
    5.25
    340
    9.65


    Solution


  49. Question

    Consider the \(y=\sqrt[3]{x}\) curve shown below (the cube root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt[3]{x}\)
    -815
    -5.68
    97
    5.73
    681
    9.68


    Solution


  50. Question

    Consider the \(y=\sqrt[3]{x}\) curve shown below (the cube root function). There are 6 points along the curve, find the missing coordinates of those points in the table.

    plot of chunk unnamed-chunk-1

    \(x\) \(y=\sqrt[3]{x}\)
    -512
    -7.55
    -323
    -5.38
    564
    8.54


    Solution


  51. Question

    Consider the \(y=\sqrt[3]{x}\) curve (cube-root function) shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 4 graphs are made by translating and reflecting the \(y=\sqrt[3]{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 4 graphs with the equations.

    1. Equation \(y=-\sqrt[3]{-(x+5)}-4\) matches graph .
    2. Equation \(y=-\sqrt[3]{x-5}+4\) matches graph .
    3. Equation \(y=-\sqrt[3]{x+5}+4\) matches graph .
    4. Equation \(y=-\sqrt[3]{-(x+5)}+4\) matches graph .


    Solution


  52. Question

    Consider the \(y=\sqrt[3]{x}\) curve (cube-root function) shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 4 graphs are made by translating and reflecting the \(y=\sqrt[3]{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 4 graphs with the equations.

    1. Equation \(y=\sqrt[3]{x-4}+5\) matches graph .
    2. Equation \(y=-\sqrt[3]{x-4}-5\) matches graph .
    3. Equation \(y=\sqrt[3]{x-4}-5\) matches graph .
    4. Equation \(y=\sqrt[3]{-(x-4)}+5\) matches graph .


    Solution


  53. Question

    Consider the \(y=\sqrt[3]{x}\) curve (cube-root function) shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 4 graphs are made by translating and reflecting the \(y=\sqrt[3]{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 4 graphs with the equations.

    1. Equation \(y=\sqrt[3]{-(x+3)}-5\) matches graph .
    2. Equation \(y=\sqrt[3]{-(x-3)}-5\) matches graph .
    3. Equation \(y=-\sqrt[3]{-(x+3)}-5\) matches graph .
    4. Equation \(y=\sqrt[3]{x-3}-5\) matches graph .


    Solution


  54. Question

    Consider the \(y=\sqrt[3]{x}\) curve (cube-root function) shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 4 graphs are made by translating and reflecting the \(y=\sqrt[3]{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 4 graphs with the equations.

    1. Equation \(y=-\sqrt[3]{x+5}-2\) matches graph .
    2. Equation \(y=\sqrt[3]{-(x+5)}+2\) matches graph .
    3. Equation \(y=-\sqrt[3]{-(x-5)}+2\) matches graph .
    4. Equation \(y=\sqrt[3]{x+5}+2\) matches graph .


    Solution


  55. Question

    Consider the \(y=\sqrt[3]{x}\) curve (cube-root function) shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 4 graphs are made by translating and reflecting the \(y=\sqrt[3]{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 4 graphs with the equations.

    1. Equation \(y=-\sqrt[3]{x-3}-2\) matches graph .
    2. Equation \(y=-\sqrt[3]{-(x+3)}+2\) matches graph .
    3. Equation \(y=-\sqrt[3]{x+3}-2\) matches graph .
    4. Equation \(y=\sqrt[3]{x-3}+2\) matches graph .


    Solution


  56. Question

    Consider the \(y=\sqrt[3]{x}\) curve (cube-root function) shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 4 graphs are made by translating and reflecting the \(y=\sqrt[3]{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 4 graphs with the equations.

    1. Equation \(y=\sqrt[3]{x-3}+5\) matches graph .
    2. Equation \(y=-\sqrt[3]{-(x+3)}-5\) matches graph .
    3. Equation \(y=\sqrt[3]{x-3}-5\) matches graph .
    4. Equation \(y=-\sqrt[3]{x+3}+5\) matches graph .


    Solution


  57. Question

    Consider the \(y=\sqrt[3]{x}\) curve (cube-root function) shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 4 graphs are made by translating and reflecting the \(y=\sqrt[3]{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 4 graphs with the equations.

    1. Equation \(y=-\sqrt[3]{x+2}+3\) matches graph .
    2. Equation \(y=\sqrt[3]{x-2}-3\) matches graph .
    3. Equation \(y=\sqrt[3]{x-2}+3\) matches graph .
    4. Equation \(y=\sqrt[3]{x+2}-3\) matches graph .


    Solution


  58. Question

    Consider the \(y=\sqrt[3]{x}\) curve (cube-root function) shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 4 graphs are made by translating and reflecting the \(y=\sqrt[3]{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 4 graphs with the equations.

    1. Equation \(y=\sqrt[3]{-(x+5)}-4\) matches graph .
    2. Equation \(y=-\sqrt[3]{-(x-5)}+4\) matches graph .
    3. Equation \(y=-\sqrt[3]{x-5}-4\) matches graph .
    4. Equation \(y=-\sqrt[3]{-(x-5)}-4\) matches graph .


    Solution


  59. Question

    Consider the \(y=\sqrt[3]{x}\) curve (cube-root function) shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 4 graphs are made by translating and reflecting the \(y=\sqrt[3]{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 4 graphs with the equations.

    1. Equation \(y=\sqrt[3]{x-2}+4\) matches graph .
    2. Equation \(y=\sqrt[3]{x+2}-4\) matches graph .
    3. Equation \(y=\sqrt[3]{-(x+2)}-4\) matches graph .
    4. Equation \(y=\sqrt[3]{-(x-2)}-4\) matches graph .


    Solution


  60. Question

    Consider the \(y=\sqrt[3]{x}\) curve (cube-root function) shown below. In the context of transforming functions, a very simple function is often called a “parent” function.

    plot of chunk unnamed-chunk-1

    The following 4 graphs are made by translating and reflecting the \(y=\sqrt[3]{x}\) curve. So each one is a different daughter function, made by altering the parent function.

    plot of chunk unnamed-chunk-2

    Match the 4 graphs with the equations.

    1. Equation \(y=-\sqrt[3]{-(x-4)}+3\) matches graph .
    2. Equation \(y=\sqrt[3]{-(x+4)}-3\) matches graph .
    3. Equation \(y=-\sqrt[3]{-(x+4)}-3\) matches graph .
    4. Equation \(y=\sqrt[3]{-(x+4)}+3\) matches graph .


    Solution


  61. Question

    Find the actual solution and the extraneous solution to the equation below.

    \[\sqrt{x+7}+1~=~x-4\]



    Solution


  62. Question

    Find the actual solution and the extraneous solution to the equation below.

    \[\sqrt{x+3}-3~=~-x-4\]



    Solution


  63. Question

    Find the actual solution and the extraneous solution to the equation below.

    \[\sqrt{x+8}+5~=~-x+3\]



    Solution


  64. Question

    Find the actual solution and the extraneous solution to the equation below.

    \[\sqrt{x+7}+6~=~-x+1\]



    Solution


  65. Question

    Find the actual solution and the extraneous solution to the equation below.

    \[\sqrt{x-3}+6~=~-x+11\]



    Solution


  66. Question

    Find the actual solution and the extraneous solution to the equation below.

    \[\sqrt{x+8}+1~=~x-3\]



    Solution


  67. Question

    Find the actual solution and the extraneous solution to the equation below.

    \[\sqrt{x-5}+8~=~-x+15\]



    Solution


  68. Question

    Find the actual solution and the extraneous solution to the equation below.

    \[\sqrt{x+7}+2~=~x+7\]



    Solution


  69. Question

    Find the actual solution and the extraneous solution to the equation below.

    \[\sqrt{x-2}+4~=~-x+8\]



    Solution


  70. Question

    Find the actual solution and the extraneous solution to the equation below.

    \[\sqrt{x-1}-8~=~x-11\]



    Solution


  71. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{(x-2)(x-5)}{(x-1)(x-5)}\]



    Solution


  72. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{(x+6)(x-1)}{(x+6)(x-2)}\]



    Solution


  73. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{(x+1)(x-3)}{(x+1)(x-4)}\]



    Solution


  74. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{(x-2)(x-3)}{(x+5)(x-2)}\]



    Solution


  75. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{(x+4)(x+3)}{(x+3)(x-2)}\]



    Solution


  76. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{(x+6)(x+4)}{(x+6)(x+5)}\]



    Solution


  77. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{(x+3)(x-2)}{(x+3)(x-6)}\]



    Solution


  78. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{(x+6)(x-5)}{(x-1)(x-5)}\]



    Solution


  79. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{(x+2)(x-3)}{(x+1)(x-3)}\]



    Solution


  80. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{(x+6)(x-1)}{(x+5)(x-1)}\]



    Solution


  81. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{x^{2}-{9}x+{18}}{x^{2}-{8}x+{12}}\]



    Solution


  82. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{x^{2}-{2}x-{24}}{x^{2}-x-{30}}\]



    Solution


  83. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{x^{2}-{5}x+{4}}{x^{2}-{6}x+{8}}\]



    Solution


  84. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{x^{2}+{6}x+{8}}{x^{2}+{10}x+{24}}\]



    Solution


  85. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{x^{2}+{2}x-{15}}{x^{2}+{6}x+{5}}\]



    Solution


  86. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{x^{2}+{2}x-{24}}{x^{2}+x-{30}}\]



    Solution


  87. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{x^{2}-{6}x+{5}}{x^{2}-{7}x+{10}}\]



    Solution


  88. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{x^{2}-{3}x-{10}}{x^{2}+x-{30}}\]



    Solution


  89. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{x^{2}+{5}x+{4}}{x^{2}-{2}x-{3}}\]



    Solution


  90. Question

    A rational function is a quotient (ratio) of two polynomials. When both polynomials are expressed in factored form, it is easy to identify horizontal positions of the \(x\)-intercepts, the removable discontinuities (also called holes), and the vertical asymptotes.

    Technically, if factors have multiplicity, it is possible for these simple rules to be “wrong”. Specifically, if the same factor is in both numerator and denominator, but with higher multiplicity in the denominator, the result is a vertical asymptote instead of a hole. This technicality is outside the scope of this class.

    The rational function below has an \(x\)-intercept, a hole, and a vertical asymptote. Identify the \(x\) coordinate of each feature.

    \[f(x)~=~\frac{x^{2}+{9}x+{20}}{x^{2}+{2}x-{15}}\]



    Solution


  91. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}-{9}x+{8}}{x^{2}-{15}x+{56}}\]

    Find the \((x,y)\)-coordinates of the hole (removable discontinuity) on the curve \(y=f(x)\).

    (, )



    Solution


  92. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}-{8}x+{12}}{x^{2}-{5}x+{6}}\]

    Find the \((x,y)\)-coordinates of the hole (removable discontinuity) on the curve \(y=f(x)\).

    (, )



    Solution


  93. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}-{9}x+{14}}{x^{2}-{15}x+{56}}\]

    Find the \((x,y)\)-coordinates of the hole (removable discontinuity) on the curve \(y=f(x)\).

    (, )



    Solution


  94. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}-{2}x-{24}}{x^{2}+{13}x+{36}}\]

    Find the \((x,y)\)-coordinates of the hole (removable discontinuity) on the curve \(y=f(x)\).

    (, )



    Solution


  95. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}-{6}x+{8}}{x^{2}-{5}x+{6}}\]

    Find the \((x,y)\)-coordinates of the hole (removable discontinuity) on the curve \(y=f(x)\).

    (, )



    Solution


  96. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}+{8}x-{20}}{x^{2}+{18}x+{80}}\]

    Find the \((x,y)\)-coordinates of the hole (removable discontinuity) on the curve \(y=f(x)\).

    (, )



    Solution


  97. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}+{2}x-{48}}{x^{2}+{18}x+{80}}\]

    Find the \((x,y)\)-coordinates of the hole (removable discontinuity) on the curve \(y=f(x)\).

    (, )



    Solution


  98. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}-{13}x+{42}}{x^{2}-{15}x+{56}}\]

    Find the \((x,y)\)-coordinates of the hole (removable discontinuity) on the curve \(y=f(x)\).

    (, )



    Solution


  99. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}+{9}x+{14}}{x^{2}+{5}x+{6}}\]

    Find the \((x,y)\)-coordinates of the hole (removable discontinuity) on the curve \(y=f(x)\).

    (, )



    Solution


  100. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}-{12}x+{20}}{x^{2}-{3}x+{2}}\]

    Find the \((x,y)\)-coordinates of the hole (removable discontinuity) on the curve \(y=f(x)\).

    (, )



    Solution


  101. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}-{4}x-{21}}{x^{2}-{15}x+{56}}\]

    Let linear function \(L\) have a slope of 1 and an unknown parameter (\(b\)) dictating the \(y\)-intercept.

    \[L(x)~=~x+B\]

    Both curves, \(y=f(x)\) and \(y=L(x)\), are graphed. The hole of function \(f\) is at point \((h,k)\), and the point \((h,k)\) lies on the line, so \(k=L(h)\). This implies that the equation \(f(x)=L(x)\) would have an extraneous solution of \(x=h\).

    Find the unknown parameter \(B\).


    Solution


  102. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}+{2}x-{35}}{x^{2}-{6}x+{5}}\]

    Let linear function \(L\) have a slope of 1 and an unknown parameter (\(b\)) dictating the \(y\)-intercept.

    \[L(x)~=~x+B\]

    Both curves, \(y=f(x)\) and \(y=L(x)\), are graphed. The hole of function \(f\) is at point \((h,k)\), and the point \((h,k)\) lies on the line, so \(k=L(h)\). This implies that the equation \(f(x)=L(x)\) would have an extraneous solution of \(x=h\).

    Find the unknown parameter \(B\).


    Solution


  103. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}-{9}x+{8}}{x^{2}-{17}x+{72}}\]

    Let linear function \(L\) have a slope of 1 and an unknown parameter (\(b\)) dictating the \(y\)-intercept.

    \[L(x)~=~x+B\]

    Both curves, \(y=f(x)\) and \(y=L(x)\), are graphed. The hole of function \(f\) is at point \((h,k)\), and the point \((h,k)\) lies on the line, so \(k=L(h)\). This implies that the equation \(f(x)=L(x)\) would have an extraneous solution of \(x=h\).

    Find the unknown parameter \(B\).


    Solution


  104. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}+{2}x-{15}}{x^{2}-{10}x+{21}}\]

    Let linear function \(L\) have a slope of 1 and an unknown parameter (\(b\)) dictating the \(y\)-intercept.

    \[L(x)~=~x+B\]

    Both curves, \(y=f(x)\) and \(y=L(x)\), are graphed. The hole of function \(f\) is at point \((h,k)\), and the point \((h,k)\) lies on the line, so \(k=L(h)\). This implies that the equation \(f(x)=L(x)\) would have an extraneous solution of \(x=h\).

    Find the unknown parameter \(B\).


    Solution


  105. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}+{18}x+{80}}{x^{2}+{19}x+{90}}\]

    Let linear function \(L\) have a slope of 1 and an unknown parameter (\(b\)) dictating the \(y\)-intercept.

    \[L(x)~=~x+B\]

    Both curves, \(y=f(x)\) and \(y=L(x)\), are graphed. The hole of function \(f\) is at point \((h,k)\), and the point \((h,k)\) lies on the line, so \(k=L(h)\). This implies that the equation \(f(x)=L(x)\) would have an extraneous solution of \(x=h\).

    Find the unknown parameter \(B\).


    Solution


  106. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}+{6}x+{5}}{x^{2}+{9}x+{20}}\]

    Let linear function \(L\) have a slope of 1 and an unknown parameter (\(b\)) dictating the \(y\)-intercept.

    \[L(x)~=~x+B\]

    Both curves, \(y=f(x)\) and \(y=L(x)\), are graphed. The hole of function \(f\) is at point \((h,k)\), and the point \((h,k)\) lies on the line, so \(k=L(h)\). This implies that the equation \(f(x)=L(x)\) would have an extraneous solution of \(x=h\).

    Find the unknown parameter \(B\).


    Solution


  107. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}+{2}x-{35}}{x^{2}+{17}x+{70}}\]

    Let linear function \(L\) have a slope of 1 and an unknown parameter (\(b\)) dictating the \(y\)-intercept.

    \[L(x)~=~x+B\]

    Both curves, \(y=f(x)\) and \(y=L(x)\), are graphed. The hole of function \(f\) is at point \((h,k)\), and the point \((h,k)\) lies on the line, so \(k=L(h)\). This implies that the equation \(f(x)=L(x)\) would have an extraneous solution of \(x=h\).

    Find the unknown parameter \(B\).


    Solution


  108. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}-x-{56}}{x^{2}-{11}x+{24}}\]

    Let linear function \(L\) have a slope of 1 and an unknown parameter (\(b\)) dictating the \(y\)-intercept.

    \[L(x)~=~x+B\]

    Both curves, \(y=f(x)\) and \(y=L(x)\), are graphed. The hole of function \(f\) is at point \((h,k)\), and the point \((h,k)\) lies on the line, so \(k=L(h)\). This implies that the equation \(f(x)=L(x)\) would have an extraneous solution of \(x=h\).

    Find the unknown parameter \(B\).


    Solution


  109. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}+{6}x-{7}}{x^{2}-{3}x+{2}}\]

    Let linear function \(L\) have a slope of 1 and an unknown parameter (\(b\)) dictating the \(y\)-intercept.

    \[L(x)~=~x+B\]

    Both curves, \(y=f(x)\) and \(y=L(x)\), are graphed. The hole of function \(f\) is at point \((h,k)\), and the point \((h,k)\) lies on the line, so \(k=L(h)\). This implies that the equation \(f(x)=L(x)\) would have an extraneous solution of \(x=h\).

    Find the unknown parameter \(B\).


    Solution


  110. Question

    Let rational function \(f\) be defined below.

    \[f(x)~=~\frac{x^{2}+{7}x+{10}}{x^{2}+{3}x+{2}}\]

    Let linear function \(L\) have a slope of 1 and an unknown parameter (\(b\)) dictating the \(y\)-intercept.

    \[L(x)~=~x+B\]

    Both curves, \(y=f(x)\) and \(y=L(x)\), are graphed. The hole of function \(f\) is at point \((h,k)\), and the point \((h,k)\) lies on the line, so \(k=L(h)\). This implies that the equation \(f(x)=L(x)\) would have an extraneous solution of \(x=h\).

    Find the unknown parameter \(B\).


    Solution


  111. Question

    The 2D distance formula is an equivalent equation to the Pythagorean equation. When finding distance between two points on an \(xy\)-plane, it can be helpful to imagine a right triangle with its legs horizontal and vertical, and its hypotenuse connecting the points. The length of the hypotenuse is the square root of the sum of the squares of the legs’ lengths. Each leg’s length is found by taking an absolute difference corresponding coordinates; for example \(|x_2-x_1|\) is the length of the horizontal leg, and \(|y_2-y_1|\) is the length of the vertical leg.

    A square field is 200 feet by 200 feet. The field is marked with a grid pattern, so the Southwest corner is at point \((0,0)\) and the Northeast corner is at point \((200,200)\). Near the middle, at position \((102, 99)\) is a beacon, marked as a large bright-red dot. On the field are 5 players; their positions are listed in the table below. Find how far each player is from the beacon.

    plot of chunk unnamed-chunk-1

    I would recommend setting this up in Desmos using \(d=\sqrt{(x_2-102)^2+(y_2-99)^2}\) and using sliders for \(x_2\) and \(y_2\).

    Player Position Distance to beacon (ft)
    1 (113,159)
    2 (67,87)
    3 (69,55)
    4 (122,84)
    5 (106,102)


    Solution


  112. Question

    The 2D distance formula is an equivalent equation to the Pythagorean equation. When finding distance between two points on an \(xy\)-plane, it can be helpful to imagine a right triangle with its legs horizontal and vertical, and its hypotenuse connecting the points. The length of the hypotenuse is the square root of the sum of the squares of the legs’ lengths. Each leg’s length is found by taking an absolute difference corresponding coordinates; for example \(|x_2-x_1|\) is the length of the horizontal leg, and \(|y_2-y_1|\) is the length of the vertical leg.

    A square field is 200 feet by 200 feet. The field is marked with a grid pattern, so the Southwest corner is at point \((0,0)\) and the Northeast corner is at point \((200,200)\). Near the middle, at position \((96, 103)\) is a beacon, marked as a large bright-red dot. On the field are 5 players; their positions are listed in the table below. Find how far each player is from the beacon.

    plot of chunk unnamed-chunk-1

    I would recommend setting this up in Desmos using \(d=\sqrt{(x_2-96)^2+(y_2-103)^2}\) and using sliders for \(x_2\) and \(y_2\).

    Player Position Distance to beacon (ft)
    1 (111,123)
    2 (156,148)
    3 (85,43)
    4 (112,73)
    5 (66,63)


    Solution


  113. Question

    The 2D distance formula is an equivalent equation to the Pythagorean equation. When finding distance between two points on an \(xy\)-plane, it can be helpful to imagine a right triangle with its legs horizontal and vertical, and its hypotenuse connecting the points. The length of the hypotenuse is the square root of the sum of the squares of the legs’ lengths. Each leg’s length is found by taking an absolute difference corresponding coordinates; for example \(|x_2-x_1|\) is the length of the horizontal leg, and \(|y_2-y_1|\) is the length of the vertical leg.

    A square field is 200 feet by 200 feet. The field is marked with a grid pattern, so the Southwest corner is at point \((0,0)\) and the Northeast corner is at point \((200,200)\). Near the middle, at position \((105, 96)\) is a beacon, marked as a large bright-red dot. On the field are 5 players; their positions are listed in the table below. Find how far each player is from the beacon.

    plot of chunk unnamed-chunk-1

    I would recommend setting this up in Desmos using \(d=\sqrt{(x_2-105)^2+(y_2-96)^2}\) and using sliders for \(x_2\) and \(y_2\).

    Player Position Distance to beacon (ft)
    1 (72,152)
    2 (81,114)
    3 (173,45)
    4 (29,39)
    5 (140,108)


    Solution


  114. Question

    The 2D distance formula is an equivalent equation to the Pythagorean equation. When finding distance between two points on an \(xy\)-plane, it can be helpful to imagine a right triangle with its legs horizontal and vertical, and its hypotenuse connecting the points. The length of the hypotenuse is the square root of the sum of the squares of the legs’ lengths. Each leg’s length is found by taking an absolute difference corresponding coordinates; for example \(|x_2-x_1|\) is the length of the horizontal leg, and \(|y_2-y_1|\) is the length of the vertical leg.

    A square field is 200 feet by 200 feet. The field is marked with a grid pattern, so the Southwest corner is at point \((0,0)\) and the Northeast corner is at point \((200,200)\). Near the middle, at position \((93, 106)\) is a beacon, marked as a large bright-red dot. On the field are 5 players; their positions are listed in the table below. Find how far each player is from the beacon.

    plot of chunk unnamed-chunk-1

    I would recommend setting this up in Desmos using \(d=\sqrt{(x_2-93)^2+(y_2-106)^2}\) and using sliders for \(x_2\) and \(y_2\).

    Player Position Distance to beacon (ft)
    1 (177,71)
    2 (77,118)
    3 (73,154)
    4 (75,26)
    5 (28,34)


    Solution


  115. Question

    The 2D distance formula is an equivalent equation to the Pythagorean equation. When finding distance between two points on an \(xy\)-plane, it can be helpful to imagine a right triangle with its legs horizontal and vertical, and its hypotenuse connecting the points. The length of the hypotenuse is the square root of the sum of the squares of the legs’ lengths. Each leg’s length is found by taking an absolute difference corresponding coordinates; for example \(|x_2-x_1|\) is the length of the horizontal leg, and \(|y_2-y_1|\) is the length of the vertical leg.

    A square field is 200 feet by 200 feet. The field is marked with a grid pattern, so the Southwest corner is at point \((0,0)\) and the Northeast corner is at point \((200,200)\). Near the middle, at position \((105, 98)\) is a beacon, marked as a large bright-red dot. On the field are 5 players; their positions are listed in the table below. Find how far each player is from the beacon.

    plot of chunk unnamed-chunk-1

    I would recommend setting this up in Desmos using \(d=\sqrt{(x_2-105)^2+(y_2-98)^2}\) and using sliders for \(x_2\) and \(y_2\).

    Player Position Distance to beacon (ft)
    1 (137,158)
    2 (63,138)
    3 (123,18)
    4 (175,122)
    5 (81,108)


    Solution


  116. Question

    The 2D distance formula is an equivalent equation to the Pythagorean equation. When finding distance between two points on an \(xy\)-plane, it can be helpful to imagine a right triangle with its legs horizontal and vertical, and its hypotenuse connecting the points. The length of the hypotenuse is the square root of the sum of the squares of the legs’ lengths. Each leg’s length is found by taking an absolute difference corresponding coordinates; for example \(|x_2-x_1|\) is the length of the horizontal leg, and \(|y_2-y_1|\) is the length of the vertical leg.

    A square field is 200 feet by 200 feet. The field is marked with a grid pattern, so the Southwest corner is at point \((0,0)\) and the Northeast corner is at point \((200,200)\). Near the middle, at position \((93, 97)\) is a beacon, marked as a large bright-red dot. On the field are 5 players; their positions are listed in the table below. Find how far each player is from the beacon.

    plot of chunk unnamed-chunk-1

    I would recommend setting this up in Desmos using \(d=\sqrt{(x_2-93)^2+(y_2-97)^2}\) and using sliders for \(x_2\) and \(y_2\).

    Player Position Distance to beacon (ft)
    1 (153,177)
    2 (72,125)
    3 (165,43)
    4 (25,148)
    5 (17,40)


    Solution


  117. Question

    The 2D distance formula is an equivalent equation to the Pythagorean equation. When finding distance between two points on an \(xy\)-plane, it can be helpful to imagine a right triangle with its legs horizontal and vertical, and its hypotenuse connecting the points. The length of the hypotenuse is the square root of the sum of the squares of the legs’ lengths. Each leg’s length is found by taking an absolute difference corresponding coordinates; for example \(|x_2-x_1|\) is the length of the horizontal leg, and \(|y_2-y_1|\) is the length of the vertical leg.

    A square field is 200 feet by 200 feet. The field is marked with a grid pattern, so the Southwest corner is at point \((0,0)\) and the Northeast corner is at point \((200,200)\). Near the middle, at position \((101, 108)\) is a beacon, marked as a large bright-red dot. On the field are 5 players; their positions are listed in the table below. Find how far each player is from the beacon.

    plot of chunk unnamed-chunk-1

    I would recommend setting this up in Desmos using \(d=\sqrt{(x_2-101)^2+(y_2-108)^2}\) and using sliders for \(x_2\) and \(y_2\).

    Player Position Distance to beacon (ft)
    1 (114,192)
    2 (146,168)
    3 (85,171)
    4 (80,128)
    5 (69,84)


    Solution


  118. Question

    The 2D distance formula is an equivalent equation to the Pythagorean equation. When finding distance between two points on an \(xy\)-plane, it can be helpful to imagine a right triangle with its legs horizontal and vertical, and its hypotenuse connecting the points. The length of the hypotenuse is the square root of the sum of the squares of the legs’ lengths. Each leg’s length is found by taking an absolute difference corresponding coordinates; for example \(|x_2-x_1|\) is the length of the horizontal leg, and \(|y_2-y_1|\) is the length of the vertical leg.

    A square field is 200 feet by 200 feet. The field is marked with a grid pattern, so the Southwest corner is at point \((0,0)\) and the Northeast corner is at point \((200,200)\). Near the middle, at position \((103, 106)\) is a beacon, marked as a large bright-red dot. On the field are 5 players; their positions are listed in the table below. Find how far each player is from the beacon.

    plot of chunk unnamed-chunk-1

    I would recommend setting this up in Desmos using \(d=\sqrt{(x_2-103)^2+(y_2-106)^2}\) and using sliders for \(x_2\) and \(y_2\).

    Player Position Distance to beacon (ft)
    1 (148,78)
    2 (121,186)
    3 (68,22)
    4 (67,121)
    5 (58,166)


    Solution


  119. Question

    The 2D distance formula is an equivalent equation to the Pythagorean equation. When finding distance between two points on an \(xy\)-plane, it can be helpful to imagine a right triangle with its legs horizontal and vertical, and its hypotenuse connecting the points. The length of the hypotenuse is the square root of the sum of the squares of the legs’ lengths. Each leg’s length is found by taking an absolute difference corresponding coordinates; for example \(|x_2-x_1|\) is the length of the horizontal leg, and \(|y_2-y_1|\) is the length of the vertical leg.

    A square field is 200 feet by 200 feet. The field is marked with a grid pattern, so the Southwest corner is at point \((0,0)\) and the Northeast corner is at point \((200,200)\). Near the middle, at position \((103, 100)\) is a beacon, marked as a large bright-red dot. On the field are 5 players; their positions are listed in the table below. Find how far each player is from the beacon.

    plot of chunk unnamed-chunk-1

    I would recommend setting this up in Desmos using \(d=\sqrt{(x_2-103)^2+(y_2-100)^2}\) and using sliders for \(x_2\) and \(y_2\).

    Player Position Distance to beacon (ft)
    1 (23,40)
    2 (75,55)
    3 (124,172)
    4 (87,112)
    5 (151,136)


    Solution


  120. Question

    The 2D distance formula is an equivalent equation to the Pythagorean equation. When finding distance between two points on an \(xy\)-plane, it can be helpful to imagine a right triangle with its legs horizontal and vertical, and its hypotenuse connecting the points. The length of the hypotenuse is the square root of the sum of the squares of the legs’ lengths. Each leg’s length is found by taking an absolute difference corresponding coordinates; for example \(|x_2-x_1|\) is the length of the horizontal leg, and \(|y_2-y_1|\) is the length of the vertical leg.

    A square field is 200 feet by 200 feet. The field is marked with a grid pattern, so the Southwest corner is at point \((0,0)\) and the Northeast corner is at point \((200,200)\). Near the middle, at position \((95, 108)\) is a beacon, marked as a large bright-red dot. On the field are 5 players; their positions are listed in the table below. Find how far each player is from the beacon.

    plot of chunk unnamed-chunk-1

    I would recommend setting this up in Desmos using \(d=\sqrt{(x_2-95)^2+(y_2-108)^2}\) and using sliders for \(x_2\) and \(y_2\).

    Player Position Distance to beacon (ft)
    1 (31,60)
    2 (147,147)
    3 (119,76)
    4 (44,40)
    5 (15,168)


    Solution


  121. Question

    A circle has a center and a radius. The distance from the center to any point on the edge equals the radius.

    In Desmos, we can graph a circle using an equation that looks like the Pythagorean equation and the 2D distance formula. If the center is at \((h,k)\) and the radius equals \(r\), the following equation graphs the desired circle.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    A large square lake is 200 miles across. The positions of 3 lighthouses are known, as well as the distance from each lighthouse to a boat.

    lighthouse position (miles, miles) distance to boat (miles)
    1 \((123,141)\) \(50\)
    2 \((9,88)\) \(85\)
    3 \((75,125)\) \(30\)

    Use trilateration to find the position of the boat:

    (, )



    Solution


  122. Question

    A circle has a center and a radius. The distance from the center to any point on the edge equals the radius.

    In Desmos, we can graph a circle using an equation that looks like the Pythagorean equation and the 2D distance formula. If the center is at \((h,k)\) and the radius equals \(r\), the following equation graphs the desired circle.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    A large square lake is 200 miles across. The positions of 3 lighthouses are known, as well as the distance from each lighthouse to a boat.

    lighthouse position (miles, miles) distance to boat (miles)
    1 \((93,102)\) \(13\)
    2 \((173,148)\) \(85\)
    3 \((9,69)\) \(100\)

    Use trilateration to find the position of the boat:

    (, )



    Solution


  123. Question

    A circle has a center and a radius. The distance from the center to any point on the edge equals the radius.

    In Desmos, we can graph a circle using an equation that looks like the Pythagorean equation and the 2D distance formula. If the center is at \((h,k)\) and the radius equals \(r\), the following equation graphs the desired circle.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    A large square lake is 200 miles across. The positions of 3 lighthouses are known, as well as the distance from each lighthouse to a boat.

    lighthouse position (miles, miles) distance to boat (miles)
    1 \((153,166)\) \(85\)
    2 \((30,44)\) \(90\)
    3 \((138,50)\) \(60\)

    Use trilateration to find the position of the boat:

    (, )



    Solution


  124. Question

    A circle has a center and a radius. The distance from the center to any point on the edge equals the radius.

    In Desmos, we can graph a circle using an equation that looks like the Pythagorean equation and the 2D distance formula. If the center is at \((h,k)\) and the radius equals \(r\), the following equation graphs the desired circle.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    A large square lake is 200 miles across. The positions of 3 lighthouses are known, as well as the distance from each lighthouse to a boat.

    lighthouse position (miles, miles) distance to boat (miles)
    1 \((66,147)\) \(50\)
    2 \((60,59)\) \(60\)
    3 \((102,99)\) \(10\)

    Use trilateration to find the position of the boat:

    (, )



    Solution


  125. Question

    A circle has a center and a radius. The distance from the center to any point on the edge equals the radius.

    In Desmos, we can graph a circle using an equation that looks like the Pythagorean equation and the 2D distance formula. If the center is at \((h,k)\) and the radius equals \(r\), the following equation graphs the desired circle.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    A large square lake is 200 miles across. The positions of 3 lighthouses are known, as well as the distance from each lighthouse to a boat.

    lighthouse position (miles, miles) distance to boat (miles)
    1 \((26,28)\) \(97\)
    2 \((73,180)\) \(82\)
    3 \((105,52)\) \(50\)

    Use trilateration to find the position of the boat:

    (, )



    Solution


  126. Question

    A circle has a center and a radius. The distance from the center to any point on the edge equals the radius.

    In Desmos, we can graph a circle using an equation that looks like the Pythagorean equation and the 2D distance formula. If the center is at \((h,k)\) and the radius equals \(r\), the following equation graphs the desired circle.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    A large square lake is 200 miles across. The positions of 3 lighthouses are known, as well as the distance from each lighthouse to a boat.

    lighthouse position (miles, miles) distance to boat (miles)
    1 \((44,160)\) \(87\)
    2 \((140,124)\) \(45\)
    3 \((168,49)\) \(80\)

    Use trilateration to find the position of the boat:

    (, )



    Solution


  127. Question

    A circle has a center and a radius. The distance from the center to any point on the edge equals the radius.

    In Desmos, we can graph a circle using an equation that looks like the Pythagorean equation and the 2D distance formula. If the center is at \((h,k)\) and the radius equals \(r\), the following equation graphs the desired circle.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    A large square lake is 200 miles across. The positions of 3 lighthouses are known, as well as the distance from each lighthouse to a boat.

    lighthouse position (miles, miles) distance to boat (miles)
    1 \((39,49)\) \(80\)
    2 \((79,129)\) \(40\)
    3 \((148,157)\) \(75\)

    Use trilateration to find the position of the boat:

    (, )



    Solution


  128. Question

    A circle has a center and a radius. The distance from the center to any point on the edge equals the radius.

    In Desmos, we can graph a circle using an equation that looks like the Pythagorean equation and the 2D distance formula. If the center is at \((h,k)\) and the radius equals \(r\), the following equation graphs the desired circle.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    A large square lake is 200 miles across. The positions of 3 lighthouses are known, as well as the distance from each lighthouse to a boat.

    lighthouse position (miles, miles) distance to boat (miles)
    1 \((81,120)\) \(35\)
    2 \((85,92)\) \(25\)
    3 \((120,39)\) \(61\)

    Use trilateration to find the position of the boat:

    (, )



    Solution


  129. Question

    A circle has a center and a radius. The distance from the center to any point on the edge equals the radius.

    In Desmos, we can graph a circle using an equation that looks like the Pythagorean equation and the 2D distance formula. If the center is at \((h,k)\) and the radius equals \(r\), the following equation graphs the desired circle.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    A large square lake is 200 miles across. The positions of 3 lighthouses are known, as well as the distance from each lighthouse to a boat.

    lighthouse position (miles, miles) distance to boat (miles)
    1 \((37,70)\) \(65\)
    2 \((139,135)\) \(58\)
    3 \((82,131)\) \(39\)

    Use trilateration to find the position of the boat:

    (, )



    Solution


  130. Question

    A circle has a center and a radius. The distance from the center to any point on the edge equals the radius.

    In Desmos, we can graph a circle using an equation that looks like the Pythagorean equation and the 2D distance formula. If the center is at \((h,k)\) and the radius equals \(r\), the following equation graphs the desired circle.

    \[(x-h)^2+(y-k)^2~=~r^2\]

    A large square lake is 200 miles across. The positions of 3 lighthouses are known, as well as the distance from each lighthouse to a boat.

    lighthouse position (miles, miles) distance to boat (miles)
    1 \((153,154)\) \(87\)
    2 \((45,71)\) \(52\)
    3 \((84,51)\) \(41\)

    Use trilateration to find the position of the boat:

    (, )



    Solution